Application:Signal Compression

Jyun-Ming Chen

Spring 2001

Signal Compression

• Lossless compression– Huffman, LZW, arithmetic, run-length– Rarely more than 2:1

• Lossy Compression– Willing to accept slight inaccuracies

• Quantization/Encoding is not discussed here                                                                

Wavelet Compression

• A function can be represented by linear combinations of any basis functions

• different bases yields different representation/approximation

Wavelet Compression (cont)

• Compression is defined by finding a smaller set of numbers to approximate the same function within the allowed error

Wavelet Compression

• : permutation of 1, …, m, then• L2 norm of approximation error

Assuming orthonormal bas

is

Wavelet Compression

• If we sort the coefficients in decreasing order, we get the desired compression (next page)

• The above computation assumes orthogonality of the basis function, which is true for most image processing wavelets

Results of Coarse Approximations (using Haar wavelets)

Significance Map

• While transmitting, an additional amount of information must be sent to indicate the positions of these significant transform values

• Either 1 or 0– Can be effectively compressed (e.g., run-length)

• Rule of thumb:– Must capture at least 99.99% of the energy to produce

acceptable approximation

Application:Denoising Signals

Types of Noise

• Random noise– Highly oscillatory– Assume the mean to be zero

• Pop noise– Occur at isolated locations

• Localized random noise– Due to short-lived disturbance in the

environment

Thresholding

• For removing random noise• Assume the following conditions hold:

– Energy of original signal is effectively captured by values greater than Ts

– Noise signal are transform values below noise threshold Tn

– Tn < Ts

• Set all transformed value less than Tn to zero

Results (Haar)• Depend on how the wavelet transform compact

the signal

Haar vs. Coif30

Choosing a Threshold Value

Transform preserves the Gaussian nature of the noise

Removing Pop andBackground Static

• See description on pp. 63-4

Types of Thresholding

Soft vs. Hard Threshold on Image Denoising

Quantitative Measure of Error

• Measure amount of error between noisy data and the original

• Aim to provide quantitative evidence for the effectiveness of noise removal

• Wavelet-based measure

signal edcontaminat:fsignal original:s

noise:nnsf

Error Measures (cont)

image original:f

imagenoisy :g

size image :,NM